OF MATHEMATICAL COMBINATORICS EDITED BY THE MADIS OF CHINESE ACADEMY OF SCIENCES March , 2011

A Smarandache-Fibonacci triple is a sequence S(n), n ≥ 0 such that S(n) = S(n − 1) + S(n − 2), where S(n) is the Smarandache function for integers n ≥ 0. Clearly, it is a generalization of Fibonacci sequence and Lucas sequence. Let G be a (p, q)-graph and {S(n)|n ≥ 0} a Smarandache-Fibonacci triple. An bijection f : V (G) → {S(0), S(1), S(2), . . . , S(q)} is said to be a super Smarandache-Fibonacci graceful graph if the induced edge labeling f(uv) = |f(u) − f(v)| is a bijection onto the set {S(1), S(2), . . . , S(q)}. Particularly, if S(n), n ≥ 0 is just the Lucas sequence, such a labeling f : V (G) → {l0, l1, l2, · · · , la} (a ǫ N) is said to be Lucas graceful labeling if the induced edge labeling f1(uv) = |f(u) − f(v)| is a bijection on to the set {l1, l2, · · · , lq}. Then G is called Lucas graceful graph if it admits Lucas graceful labeling. Also an injective function f : V (G) → {l0, l1, l2, · · · , lq} is said to be strong Lucas graceful labeling if the induced edge labeling f1(uv) = |f(u) − f(v)| is a bijection onto the set {l1, l2, ..., lq}. G is called strong Lucas graceful graph if it admits strong Lucas graceful labeling. In this paper, we show that some graphs namely Pn, P + n − e, Sm,n, Fm@Pn, Cm@Pn, K1,n ⊙ 2Pm, C3@2Pn and Cn@K1,2 admit Lucas graceful labeling and some graphs namely K1,n and Fn admit strong Lucas graceful labeling.